Hawking Temperature for Near-Equilibrium Black Holes

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In the AdS/CFT setup, if we were to define temperature at a point on the boundary in this way, we would need information in the bulk region, which is not causally accessible from that boundary point. Such a property would not be desirable if we care about causality in the bulk and boundary, especially when time dependence comes into the story. To avoid such problems, we consider the conventional derivation of the Hawking radiation based on Bogoliubov transformations for nonstationary background. An advantage of this method is that we can naturally determine temperature measured by asymptotic observers using only information causally available to them.1 Particularly, we focus our attention on an eternal black hole rather than black hole formation by gravitational collapse because we are interested in near-equilibrium system such as transition from an equilibrium state to another equilibrium state.2 In other words, we focus on late-time dynamics after the black hole has formed. We clarify how the Hawking temperature changes when a black hole becomes dynamical, for instance, due to mass accretion to the black hole, and show that the Hawking temperature of dynamical black holes can be naturally associated with “surface gravity” of the past horizon. The paper is organized as follows. In Sec. II, we evaluate the Bogoliubov coefficients using the saddle point approximation and show that the “surface gravity” of the past horizon gives the Hawking temperature observed in the asymptotic region. In Sec. III, as illustrations of the use of our method, we consider applications to simple examples in asymptotically flat and AdS cases. We summarize and discuss implications of the results in Sec. IV.

II. ESTIMATION OF BOGOLIUBOV COEFFICIENTS

We estimate Bogoliubov coefficients in order to define Hawking temperature for a nonstationary background in this section. We consider an eternal black hole in thermal equilibrium with its surroundings, that is, in the Hartle–Hawking state. In asymptotically flat cases, we should immerse the black hole into a thermal bath to realize this state. In asymptotically AdS cases, on the other hand, it is naturally realized due to its boundary condition [26, 27] because the black hole is enclosed in the AdS boundary. We exploit the saddle point approximation to evaluate the Bogoliubov coefficients and determine the temperature from them. After reviewing usage of the approximation for the static spacetime in Sec. II A, we consider its extension to nonstationary spacetime in Sec. II B. In this extension to nonstationary spacetime, we introduce a quantity, κ(u), which may be interpreted as an extension of the surface gravity of static Killing horizon. In Sec. II C, we reinterpret this quantity from a geometric point of view, and clarify that κ(u) is naturally associated with the past horizon of the eternal black hole.

A. Static spacetime and the saddle point approximation

For simplicity, we consider the two-dimensional part of the spacetime consisting of time and radial directions, that is, we focus on the s-wave sector of radiation. We also adopt the geometric optics approximation, and neglect the backscattering of the waves due to the curvature of spacetime. We introduce the null coordinate u which gives a natural time for observers in the asymptotic region (null infinity). If the spacetime is stationary, this time corresponds to the Killing time. We introduce another null coordinate U which is the affine parameter on the past horizon. For the stationary case it becomes the familiar Kruskal coordinate. Because lines described by u = const. are outgoing null geodesics, geodesic equations give a relation U = U(u) between the two null coordinates. Now, we consider a massless scalar field.3 In the current setup, solutions of the field equation are simply given by arbitrary functions of each null coordinate. In a standard manner we can define positive frequency modes with −iωu respect to u and U, respectively. Then, the Bogoliubov transformation between two sets of modes uω e and u¯ e−iωUˆ is determined by the Bogoliubov coefficients ∝ ωˆ ∝ ∞ αωωˆ i ωˆ dU = du e±iωu−iωUˆ (u), (1) β ±2π ω du ωωˆ r Z−∞

1 The essential part of the calculation is similar to Ref. [9]. 2 The derivations are quite similar even on the background spacetime with gravitational collapse and black hole formation. See also comments in Sec. IV. 3 In asymptotically AdS cases we should consider a conformally coupled scalar ﬁeld. 3 where the upper and lower signs correspond to αωωˆ and βωωˆ , respectively. Now, to evaluate those coeﬃcients by the saddle point approximation, we will consider the integral

∞ exp φ(u)du, (2) Z−∞ where dU φ(u) log iωu iωUˆ (u). (3) ≡ du ± − ′ Saddle points are located at u = u∗ given by φ (u∗) = 0, where

d dU dU φ′(u)= log iω iωˆ , (4) du du ± − du and a prime denotes a derivative. Consider a static black hole, for example. When the surface gravity at the Killing horizon is κ, the relation between the two coordinates u and U is given by U(u)= exp( κu), which is the coordinate transformation for the Kruskal coordinate. Then, we have − −

′ −κu∗ φ (u∗)= κ iω iωκeˆ =0, (5) − ± − and the saddle point

∓iθ −1 ire 1 r i π u∗ = κ log = log θ , (6) − ωˆ −κ ωˆ − κ 2 ∓ ∓iθ where re 1 iω/κ. We notice that the real part of u∗ depends onω ˆ and it becomes larger asω ˆ increases. This implies≡ that∓ the Kruskal modes with very high frequenciesω ˆ are relevant for late-time features, while relevant frequenciesω ˆ will change depending on the time of observation. [See also discussions around Eq. (30).] Using

−1∓iθ ∓iθ ire φ(u∗) = log κ + re log , ωˆ (7) ′′ 2 ∓iθ φ (u∗)= κ re , − we have the saddle point approximation of the integral (2) as

re∓θ ∞ ∞ ′′ −1∓iθ φ (u∗) 2 2π ire 2π φ(u) φ(u∗) 2 (u−u∗) φ(u∗) e du e e du = e ′′ = κ ∓iθ , (8) ≃ φ (u∗) ωˆ re Z−∞ Z−∞ s− r which becomes a good approximation when ω/κ 1. This expression derives the Bogoliubov coeﬃcients of Eq. (1) given by ≫

α 2 2πω | ωωˆ | exp . (9) β 2 ≃ κ | ωωˆ | Thus, we have the familiar result of the Hawking temperature to be T = κ/2π by the saddle point approximation. We could also perform the integral explicitly to obtain a well-known exact expression [1]

∞ ∓iθ eφ(u)du = κ( iωˆ)−re Γ re∓iθ . (10) − Z−∞ Using the Stirling formula of the gamma function z z Γ(1 + z)= zΓ(z) √2πz , (11) ≃ e we have Eq. (8) again from Eq. (10). This result indicates that when the geometric optics approximation is satisﬁed, namely, when ω/κ 1, the saddle point approximation is valid. ≫ 4

B. Extension to nonstationary spacetime

Let us now take a nonstationary spacetime background. To probe this spacetime, we use wave packets which are localized in both the time and frequency domains rather than plane waves spreading over the whole time. The wave packet is peaked around a time u = u0 with width ∆u, which are the time and duration of the observation, respectively. The Bogoliubov coeﬃcients for a wave packet∼ are obtained by inserting a window function into the integrand of Eq. (1), that is,

∞ Aωωˆ 1 ωˆ dU = w (u u ) e±iωu−iωUˆ (u)du, (12) B ±2π ω ∆u − 0 du ωωˆ r Z−∞ where w∆u(x) is a window function which goes suﬃciently fast to zero outside the interval ∆u around x = 0. In addition, we assume that analytic continuation of w∆u(x) is varying slowly at least for x < ∆u in the complex plane. Using the Klein–Gordon product, these coeﬃcients are written as A = (ψ , u¯ ) and| |B = (ψ∗ , u¯ ), where we ωωˆ ω ωˆ ωωˆ − ω ωˆ have constructed the wave packet localized around time u = u0 and frequency ω as

1 −iωu ψω(u)= w∆u(u u0)e . (13) √4πω −

We note that Aωωˆ and Bωωˆ satisfy the following relation ∞ dˆω A 2 B 2 = (ψ , ψ ), (14) | ωωˆ | − | ωωˆ | ω ω Z0 which follows from the completeness of u¯ . The number density in terms of the wave packet mode ψ is given by { ωˆ } ω 1 ∞ dˆω B 2, (15) (ψ , ψ ) | ωωˆ | ω ω Z0 which is roughly the number of particles generated in the frequency band of width 1/∆u around frequency ω. ∼ Let us suppose that the evolution of the background spacetime is suﬃciently slow around u = u0 within time interval ∆u. To clarify what slowly evolving is, we shall introduce the following quantity d dU κ(u) log . (16) ≡−du du If the background spacetime is stationary, κ(u) becomes a constant and it is nothing but the surface gravity of the black hole (Killing) horizon, as we have seen in the previous section. Therefore, we may state that the spacetime is evolving slowly when κ(u) is almost constant during ∆u. In that case, we may express κ(u) as

κ(u)= κ0 [1 + ǫf(u)] , (17) where f(u) is a real function which satisﬁes f(u0) = 0 and f(u) 1 for u u0 < ∆u, and ǫ is a small parameter. In other words, this assumption implies | |≤ | − | ∆κ ǫ for u u0 < ∆u, (18) κ0 ≤ | − | where ∆κ κ(u) κ . For discussions below, we further assume that analytic continuation of f(u) satisﬁes f(u) 1 ≡ | − 0| | |≤ for any complex u such that u u0 < ∆u. Then, we may expand U ′(u|),−U(u|) and U ′′(u) with respect to ǫ as

u U ′(u)= U ′ exp κ(x)dx = U ′ exp κ (δu + ǫg(u)) = U ′ e−κ0δu 1 ǫκ g(u)+ (ǫ2) , 0 − 0 {− 0 } 0 − 0 O Zu0 u 1 e−κ0δu u (19) U(u)= U + U ′(x)dx = U + U ′ − ǫκ e−κ0(x−u0)g(x)dx + (ǫ2) , 0 0 0 κ − 0 O Zu0 0 Zu0 U ′′(u)= κ(u)U ′(u)= κ (1 + ǫf(u)) U ′ e−κ0δu 1 ǫκ g(u)+ (ǫ2) , − − 0 0 − 0 O where we deﬁned U = U(u ), δu = u u and 0 0 − 0 u g(u)= f(x)dx. (20) Zu0 5

Note that g(u) δu follows from the assumption f(u) 1 for δu < ∆u. We also expand the saddle point u = u∗ as | | ≤ | | | |≤ | |

(0) (1) 2 u∗ = u∗ + ǫu∗ + (ǫ ). (21) O Then, the equation for the saddle point from Eq. (4) is expanded using Eq. (19) as

(0) (0) ′ ∓iθ ′ −κ0δu (0) ′ −κ0δu (1) (0) 2 0= φ (u∗)= κ re iωUˆ e ∗ + ǫκ f u∗ + iωUˆ e ∗ u∗ + g u∗ + (ǫ ), (22) − 0 − 0 0 − 0 O h i ∓iθ where we have redefined re 1 iω/κ0, which satisfies r 1 because of the geometric optics approximation ′ ≡ ∓ ≫ ω/κ 1. Solving φ (u∗) = 0 order by order of ǫ, we find 0 ≫ ∓iθ (0) −1 iκ0re 1 κ0r i π δu∗ = κ log = log θ , − 0 ωUˆ ′ −κ ωUˆ ′ − κ 2 ∓ 0 0 0 0 (0) (0) (23) κ0δu (1) (0) e ∗ (0) (0) f u∗ u∗ = g u∗ + ′ f u∗ = g u∗ ∓iθ . − iωUˆ 0 − − κ0re ′′ ′ ′′ Plugging Eq. (23) into Eq. (3) and φ (u) = κ (u) iωUˆ (u) at u = u∗ and expanding with respect to ǫ, we find ′′ ′′ − ′′ − ′ φ(u∗) φ(u ) φ + ǫφ and φ (u∗) φ + ǫφ are given by φ(u ) log U iωu iωUˆ , − 0 ≡ 0 1 ≡ 0 1 0 ≡ 0 ± 0 − 0 (0) ′ u∗ ∓iθ (0) iωUˆ 0 (0) ′ −κ0δu φ = re κ δu∗ +1 , φ = κ g u∗ + iωUˆ κ e g(u)du (24) 0 − 0 − κ 1 − 0 0 0 0 Zu0 and

′′ 2 ∓iθ ′′ ′ (0) ∓iθ (0) φ = κ re , φ = κ f u∗ + κ 1+ re f u∗ , (25) 0 − 0 1 − 0 0 where (ǫ2) terms are omitted. To guaranteeO that the perturbative expansion above is valid and that the calculation of the saddle point approxima- (1) ′′ tion is not aﬀected by the correction terms, we should require each of correction terms [ǫu∗ , ǫφ1 and ǫφ1 in Eqs. (23), (0) ′′ (1) (0) (24) and (25)] to be much smaller than their leading term (δu∗ , φ0 and φ0 , respectively). As for ǫ u∗ δu∗ ′′ ′′ ′ (0) ≪ and ǫ φ φ , we can see from Eqs. (23) and (25) that we should assume both ǫ 1 and f u∗ κ r to 1 ≪ 0 ≪ ≪ 0 hold to satisfy such requirements, where we used g(u) δu . As for ǫ Re φ1 Re φ0 , we have | | ≤ | | | | ≪ | | (0) (0) Re φ = ωIm u∗ κ Re δu∗ 1, 0 ∓ − 0 − (0) (0) Re u∗ +iIm u∗ (26) (0) ′ −κ0δu Re φ1 = κ0Re g u∗ + Re iωUˆ 0κ0 e g(u)du , − (0) ZRe u∗ ! and

(0) 2 (0) (0) Re φ <κ δu∗ + κ r δu∗ Im u∗ . (27) | 1| 0 0 (0) Then, for ǫ Re φ1 Re φ0 to hold, we need to additionally require ǫκ0 δu ∗ 1. Under these assumptions, we ﬁnd that the| integral| ≪ in| Eq.| (12) can be evaluated by the saddle point approximation≪ as

∞ φ(u) ′ (0) Re φ0 2π w (δu)e du U w δu∗ e . (28) ∆u ≃ 0 ∆u φ′′ Z−∞ s 0 | |

So that the suppression due to the window function, w∆u, is weak and the integral value becomes close to that for (0) the unwindowed function, the saddle point should satisfy δu∗ . ∆u. For the imaginary part of u∗ the condition | | δu∗ . ∆u becomes | | 1 . κ , (29) ∆u 0 and for the real part it becomes a condition onω ˆ, given by

κ0r −κ0∆u κ0r κ0∆u ′ e . ωˆ . ′ e . (30) U0 U0 6

Ifω ˆ is out of this region, no saddle point exists between u u0 . ∆u, and then the integral will be suppressed due to the window function. This condition means that only the| − Kruskal| modes ofω ˆ with limited frequency band, speciﬁed by Eq. (30), can have correlations with the wave packet mode ψω due to its localization in the time domain. Consequently, suﬃcient conditions for the saddle point approximation to be valid are given by 1 ǫκ . κ ω, f ′(u) κ r, (31) 0 ≪ ∆u 0 ≪ ≪ 0

−1 for u u0 < ∆u. Note that the condition on ǫ, ǫκ0 ∆u , can be rewritten as ∆κ∆u 1, where ∆κ is defined | − | −≪1 ≪ by Eq. (18). If we can take the time interval ∆u(& κ0 ) satisfying this condition, we have a sufficiently small ǫ 1 for which Eq. (31) holds. Roughly speaking, this condition is indicating that the time variation of κ(u) should≪ be 1 dκ −1 sufficiently moderate to satisfy κ2 du 1 between the interval u u0 . κ0 . Under these assumptions, at the 0 ≪ | − | leading order we have

A 2 2πω | ωωˆ | exp , (32) B 2 ≃ κ | ωωˆ | 0 which implies the spectrum observed at the time u = u0 becomes a thermal one with temperature κ0/2π for high frequencies ω κ . ≫ 0

C. Surface gravity for past horizon

In this section, we consider the geometrical meaning of κ(u) deﬁned by Eq. (16) in the previous section. Ingoing null vectors with respect to null coordinates u and U are written as

∂ a ∂ a ka = , k¯a = , (33) ∂u ∂U respectively. Note that U is an aﬃne parameter on the past horizon because it is one of the Kruskal coordinates. Using ka = U ′(u)k¯a, we have

ka kb = ka (U ′(u)k¯b) = (ka U ′(u))k¯b + (U ′)2k¯a k¯b. (34) ∇a ∇a ∇a ∇a The last term vanishes because U is the affine parameter. As a result, we obtain ka kb = κ(u)kb, where ∇a − d dU κ(u)= log . (35) −du du We shall call κ(u) “surface gravity” for the past horizon because it describes the inaffinity of the null generator ka of the past horizon, which is defined by the asymptotic time at the null infinity. When the spacetime is stationary, the past horizon and κ(u) coincide with the Killing horizon and its surface gravity, respectively. In this sense, κ(u) is a natural extension of surface gravity of stationary spacetimes. (In Ref. [9], it was called “peeling properties” for gravitational collapse.) It is worth noting that the surface gravity for the past horizon is not determined only by local geometrical quantities on that horizon but also by the relation between the time coordinates on the horizon and in the asymptotic region. In fact, the Killing vector which defines the surface gravity of the Killing horizon is also determined using the asymptotic time in a stationary case. Roughly speaking, the particles of the Hawking radiation observed in the asymptotic region start from the vicinity of the horizon and propagate along outgoing null geodesics. Every outgoing null ray arriving at the null infinity expe- riences the near-horizon region of the past horizon rather than the (future) event horizon. Based on this observation, it seems to be natural that the spectrum is affected by the surface gravity near the past horizon.

III. APPLICATIONS

In this section, we consider explicit examples of transitions from an initial equilibrium state to another equilibrium state. First, as a simplest example, we consider null shell accretion into a black hole in asymptotically ﬂat spacetime. Next, we will focus attention on the AdS–Vaidya spacetime. This is a toy model to describe a thermalization process in the AdS/CFT correspondence. 7

A. Asymptotically ﬂat case: null shell accretion

In this section, we consider a black hole in asymptotically flat spacetime, whose mass is initially Mi and changes into the final mass Mf due to accretion of a null shell. Then, the Hawking temperature should initially be that of the initial black hole and eventually become the temperature of the final one. The metric of a static and spherically symmetric black hole solution is given by 2 2 2 dr 2 2 ds = fI(r)dt + + r dΩ , (36) − fI(r) where the subscript I is i or f, which are for initial and final quantities, respectively. Double-null coordinates (u, v) are given by dr v u =2 , u + v =2t (37) − f(r) Z To describe null shell accretion, we will join two spacetimes at v = 0 in a manner such that for v < 0 we use the metric of the initial black hole and for v > 0 that of the final one. At v = 0 the junction condition implies that the radial coordinate should be identified as dr dr u = 2 , u = 2 . (38) i − f (r) f − f (r) Z i Z f Also, it leads to the following relations dR f (R) dR f (R) = i , = f , (39) dui − 2 duf − 2 where R is the radius of the null shell, and these are nothing but the equations of motion for the null shell. The Kruskal coordinate at the past horizon, namely, that of the initial black hole, is U = exp( κ u ), (40) − − i i where κi denotes the surface gravity of the initial black hole, and the retarded time at null infinity is that of the final black hole as u = uf .

dU dui ff (R) = κi exp( κiui) = κi exp( κiui) (41) du − du − fi(R) The surface gravity of the past horizon is ′ ′ d dU dui ff (R) dR fi (R) dR κ(u)= log = κi + − du du du − ff (R) du fi(R) du (42) f ′(R) f (R) f ′(R) = κ i f + f , i − 2 f (R) 2 i where R(u) denotes the radius of the null shell at time u. For early time u , the radius of the shell becomes → −∞ R and we have κ(u) κi. For late time u , the radius of the shell approaches the horizon radius of the → ∞ → → ∞ ′ ﬁnal black hole R rf and we have κ(u) κf as u , where we have used κf = ff (rf )/2. These imply that the asymptotic observers→ detect a change of the→ Hawking→ radiation ∞ at the retarded time when the null shell comes into the vicinity of the black hole horizon. Consider the four-dimensional Schwarzschild case with f (r)=1 2M /r for example. We have I − I R(u) 2∆M d ∆M(R(u) 2Mf) κ(u)= − , κ(u)= − 3 , (43) 4MiR(u) du − 4MiR(u) where ∆M M M . If M ∆M, the condition ≡ f − i i ≫ 1 dκ ∆M 2 . 1 (44) κ du Mi ≪ is satisﬁed for any retarded time u. Therefore, it is concluded that the Hawking temperature observed by asymptotic observers at time u is given by κ(u)/2π in the current case. We note that κ(u) changes gradually even though the spacetime describing the null shell accretion is not smooth. 8

B. Asymptotically anti-de Sitter case: AdS–Vaidya

Next, we discuss an asymptotically anti-de Sitter background. In this case, null inﬁnity at which the Hawking temperature would be observed is not a null surface but a timelike surface, namely, the so-called AdS boundary. Moreover, even when there is no black hole horizon, asymptotically AdS spacetime has a past horizon (not a white hole horizon but a Cauchy horizon). We consider the 5D AdS–Vaidya spacetime 1 ds2 = [ F (¯v,z)d¯v2 2d¯vdz + d~x2], (45) z2 − − 3 where F (¯v,z) is given by

F (¯v,z)=1 2m(¯v)z4, (46) − and the curvature radius of the AdS is set to unity. We suppose that the mass function m(¯v) is

m0 (¯v < 0)

2 πv¯ m(¯v)=  m0 + ∆m sin (0 v¯ ∆v) . (47)  2∆v ≤ ≤  m0 + ∆m (¯v > ∆v)  During an interval ∆v, the null ﬂuid is injected into the black hole with mass m0, and the mass eventually becomes m0 + ∆m. For the AdS–Vaidya, Bondi energy observed at the boundary is described by the mass function m(¯v). Note that the time coordinatev ¯ is nothing but asymptotic time at the boundary. In the context of the AdS/CFT correspondence, m(¯v) corresponds to the energy density of the CFT matter. Therefore, the duration of the injection ∆v will represent the time scale of energy density change. Now we shall consider the Hawking temperature in the current case. We introduce double-null coordinates (u, v) as

z = z(u, v), v¯ = v. (48)

In these coordinates, the boundary z = 0 is described by u = v. An asymptotic time t at the boundary is given by u+v t = 2 , and it is equal to v (or u) there. The coordinate condition for (u, v) to be the double-null coordinates leads to ∂z F (¯v,z) = , (49) ∂v − 2 which is equivalent to the geodesic equation for the outgoing null ray described by u-constant line. Note that we have ∂z ∂z 1 ∂v z=0 = ∂u z=0 = 2 as the boundary condition at z = 0. In order− to calculate− the surface gravity for the past horizon we need to know the relation between the asymptotic time and the affine parameter at the past horizon. Because the spacetime is static (strictly speaking, independent of v¯) for the initial black hole region (¯v < 0), the canonical null coordinateu ¯ is given by 2 d¯u = dv¯ + dz, (50) F where we note that the metric function F depends only on z. Moreover, the affine parameter at the past horizon, namely, the Kruskal coordinate U is immediately given by U(¯u) = exp( κ u¯), where κ is the surface gravity − − i i determined by the initial black hole with the mass m0. It is worth noting that, in general, the above null coordinate u¯ is different from the asymptotic time u defined previously. Let us recall the definition of u. The differential form becomes ∂z ∂z F dz = du + dv = ( λdu + dv), (51) ∂u ∂v − 2 − where λ is defined by 2 ∂z d¯u λ(u) = . (52) ≡ F ∂u du 9

4 10 ∆v=0.1 ∆v=0.2 ∆v=0.5 ∆ 1 3.5 v=1 ∆v=2

) 0.1 u ) ( u κ

( 3 κ - f

κ 0.01 ∆v=0.1 2.5 ∆ =0.2 0.001 v ∆v=0.5 ∆v=1 ∆v=2 2 0.0001 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 Boundary time (u=v) Boundary time (u=v)

FIG. 1: κ(u) as a function of time (u = v) at the boundary (left) and behaviors of κ(u) near the ﬁnal temperature κf (right). Time scales of injection are taken to be ∆v = 0.1, 0.2, 0.5, 1, 2, which are shorter than a typical time scale given by temperature. We set m0 = 1/2 and ∆m = 15/2, which give κi = 2 and κf = 4, respectively.

If the spacetime is static, we can take the canonical double-null form such that λ = 1 everywhere. It turns out that λ describes the relation between the asymptotic time u at the boundary and the canonical timeu ¯ in the initial black hole region, that is, the redshift factor for the outgoing null ray which goes from the initial black hole region to the boundary, due to dynamical background. (See the Appendix for details.) Now, we can obtain λ(u) as follows. Diﬀerentiating Eq. (49) with respect to u, we have

∂ ∂z 1 ∂F ∂z = . (53) ∂v ∂u −2 ∂z ∂u Integrating the above equation and the geodesic equation with respect to v from the boundary v = u to the initial ∂z static region (v < 0), we obtain ∂u for v < 0. Note that we must integrate those to the past horizon (v = ) in general. However, in the current case it is enough to evaluate only up to v = 0, because the spacetime becomes−∞ static for v< 0 and the redshift factor λ(u) does not change any more. As a result, we have d dU d κ(u)= log = κ λ(u) log λ(u), (54) −du du i − du

dU dU where we have used du = λ d¯u . In Fig. 1 we plot κ(u) as a function of the boundary time (u = v) for shorter injection times ∆v =0.1, 0.2, 0.5, 1, 2. The initial and final temperatures are given by κi = 2 and κf = 4, respectively. The conditions of Eq. (31) are almost satisfied for these parameters and thus the results of the saddle point approximation are valid.4 In these cases, we find that κ(u) converges into the final temperature exponentially after a transient phase. Time scales of the whole process of temperature change are roughly given by 3 and do not strongly depend on time scales of injection but are dominated by exponential relaxation to the final∼ temperature. Recalling that the time scale determined by temperature T of a black hole is given by 1/T =2π/κ, we may interpret this result as the time scale of the temperature change is governed by the temperature of the final black hole. See also Fig. 2 about final-temperature dependence of relaxation rate for the case of null shell accretion, for which the injection is instantaneous and equivalent to the case of ∆v = 0. Note that since we are focusing on time scales shorter than the thermal time scale ∆u κ−1 and the minimum time resolution of observation is given by this thermal time scale, we should interpret κ(u)∼ not as a value at time u but rather a time-averaged value over time scale κ−1. As the injection time becomes longer than the thermal time∼ scale, κ(u) begins to show qualitatively new behaviors. 1/4 In Fig. 3, for ∆v = 0, 2, 10, 50 we plot κ(u) together with a quasistatic temperature 2πTqs = 2(2m(v)) , which is

4 If the difference between the initial and final temperatures becomes very large, the adiabatic condition may be temporarily violated. This means that the system is so far from equilibrium that temperature can not be defined at that time. However, after the system has relaxed near the final temperature, the interpretation of κ(u) as temperature becomes well-defined again. 10

100 κ f=4 κ =8 κ f 10 f=16

1 ) u (

κ 0.1 - f κ 0.01

0.001

0.0001 0 1 2 3 4 5 Boundary time (u=v)

FIG. 2: Relaxation rate for ﬁnal temperatures κf = 4, 8, 16 for the case of null shell accretion (∆v = 0). The initial temperature is given by κi = 2. In any case, an exponential decay phase starts after a transient phase, and the decay rate in the exponential decay phase is proportional to inverse of the ﬁnal temperature.

4 4

3.5 3.5

3 3 Temperature Temperature 2.5 2.5

κ(u) κ(u) 2 2(2m(v))1/4 2 2(2m(v))1/4 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 Boundary time (u=v) Boundary time (u=v) (a)∆v = 0 (b)∆v = 2

4 4

3.5 3.5

3 3 Temperature Temperature 2.5 2.5

κ(u) κ(u) 2 2(2m(v))1/4 2 2(2m(v))1/4 0 5 10 15 20 0 20 40 60 80 100 Boundary time (u=v) Boundary time (u=v) (c)∆v = 10 (d)∆v = 50

1/4 FIG. 3: κ(u) and a quasistatic temperature 2πTqs = 2(2m(v)) for various injection times ∆v.

naively determined from the mass functions m(v) in a similar manner to static black holes. We can observe that both curves begin to coincide as the injection time becomes longer. In other words, the variation of the temperature κ(u) tends to follow the variation of the black hole mass m(v). These results lead us to the conclusion that, even if the mass injection is rapid, the temperature will gradually change over a time scale no shorter than the thermal time scale κ−1. In this sense, we may say that there is a ﬁnite relaxation time irrespective of the speed of energy injection. When∼ energy injection and the resultant black hole evolution is slower than this relaxation time, the temperature tends to evolve in a similar way to the mass injection. 11

IV. SUMMARY AND DISCUSSION

In this paper we have discussed Hawking temperature for nonstationary spacetimes. We introduced a useful measure for time evolution of spacetime, and showed that the Hawking temperature at each time is determined by the surface gravity of the past horizon when the time evolution of the spacetime is sufficiently slow. This definition of temperature is quite natural because observers can determine it without using causally inaccessible information, such as the position of the event horizon. Although we have considered an eternal black hole which has the past horizon, the essential calculation does not change for black hole formation in an asymptotically flat spacetime. In this case we should draw null rays from future null infinity back to the past null infinity instead of the past horizon, and as a result we can reproduce the original derivation of the Hawking radiation. The definition of κ(u) is the same as Eq. (16), even though it cannot be interpreted as “surface gravity” and should be understood rather in the context of “peeling properties” of outgoing null geodesics, which was discussed in [9]. It would be interesting to study the properties of κ(u) for black holes in an expanding universe, such as those discussed in Refs. [5, 28], for which the derivation is similar to the case of asymptotically flat spacetime. In the case of asymptotically AdS spacetime, in the Poincaréchart the past horizon, which is the Cauchy horizon, exists even in the pure AdS spacetime. This setup is quite similar to the eternal black hole, and we can naturally interpret κ(u) as the “surface gravity” of this past horizon. In the context of the AdS/CFT correspondence, our results indicate that quantum fluctuation of the boundary CFT is governed by the surface gravity of the past horizon when the bulk is the eternal black hole, namely, the thermal equilibrium state. Particularly, our result can be applied to phenomena discussed in Refs. [24, 25], which are directly related to the Hawking radiation in the bulk. It may be interesting to study its implications further. It would be also interesting to study κ(u) in numerical solutions obtained in, e.g., Refs. [14, 15, 23], though in general numerical calculation of κ(u) becomes difficult at late time due to exponential pileup of outgoing null geodesics onto the horizon. It is shown that the change of the Hawking temperature is delayed by thermal time scale compared with the mass injection from the boundary. This behavior can be interpreted as follows: The injected matter falls down into the black hole and changes the geometry in the vicinity of the horizon. After that, the Hawking radiation affected by this change emanates from there and reaches the boundary. This process of infall of the matter and return of the modified radiation will take finite time and, roughly speaking, this is the origin of the delay we observed. From the viewpoint of the boundary theory, this delay time can be interpreted as the relaxation time scale needed to achieve thermal equilibrium after the energy injection. For planar AdS black holes (namely, large black holes in asymptotically AdS spacetime) the horizon is located relatively near to the AdS boundary. If a black hole evolves sufficiently slower than the delay time, we may neglect its delay. In that case, the quasistatic approximation, such that one synchronizes dynamics of the event or apparent horizon and the boundary in terms of the advanced time and uses quantities associated with the future horizon, might be justified. What we emphasize is that the “surface gravity” of the past horizon, κ(u), is not only a geometrical quantity but also one with a physical meaning in the sense that it governs the thermal spectrum of the Hawking radiation observed by asymptotic observers. Many of thermodynamic properties of black holes, however, tend to be associated with the future event or apparent horizon in the previous works, not with the past horizon. It is open to discussion how these points of view are related to each other. It may be also fruitful to study how κ(u) is related to other probes of the black hole spacetime, such as those discussed in Ref. [22], and how our approach is related to other derivations of the Hawking radiation, such as the tunneling approach.

Acknowledgments

We would like to thank Takahiro Tanaka for helpful discussions. We would also like to thank Nemanja Kaloper, McCullen Sandora, Ken-ichi Nakao, Shinji Mukohyama and Tadashi Takayanagi for valuable comments. N.T. ac- knowledges hospitality at the Centro de Ciencias de Benasque during the Strings and Gravity Workshop, and thanks the participants for useful discussions. S.K. is supported by JSPS Grant-in-Aid for Creative Scientiﬁc Research No. 19GS0219. N.T. is supported in part by the DOE Grant DE-FG03-91ER40674. 12

Appendix A: redshift factor

In this appendix we show that λ(u) deﬁned by Eq. (52) is the redshift factor for the outgoing null ray. Now, we recall that the two-dimensional part of the metric is given by 1 ds2 = [ F (¯v,z)d¯v2 2d¯vdz], (A1) z2 − − A coordinate transformation such as z = z(u, v)andv ¯ = v to the double-null coordinates (u, v) gives us the diﬀerential form ∂z ∂z F dz = du + dv = ( λdu + dv), (A2) ∂u ∂v − 2 −

∂z ∂z where we have defined as λ ∂u ∂v . We consider a timelike vector≡− field ξa = ∂/∂v¯ which characterizes a natural Killing time in stationary regions and also in asymptotic regions near the boundary. Note that frequencies of the Killing modes are defined with respect to this Killing time. By using the 1-form it is rewritten as 1 ξ = (F d¯v + dz). (A3) a −z2 The tangent vector of outgoing null geodesics is given by d la , (A4) ≡ ds where s is an affine parameter, and then the geodesic equations lead to

d v¯˙ ∂F v¯˙ 2 F = , z˙ = v,¯˙ (A5) ds z2 ∂z 2z2 − 2 where the dot denotes the derivative with respect to s. We note that the last equation is equivalent to the coordinate condition ∂z/∂v = F/2 deﬁning the double-null coordinates. The “Killing energy”− E associated with the null geodesics is deﬁned by

1 F v¯˙ E laξ = (F v¯˙ +z ˙)= . (A6) ≡− a z2 2 z2 If ξa is truly a Killing ﬁeld, E should be constant along the null geodesics. From the geodesic equations we have

d v¯˙ 1 ∂F v¯˙ = , (A7) d¯v z2 2 ∂z z2 where we have used d/ds = v¯˙ d/d¯v. On the other hand, diﬀerentiating ∂z/∂v = F/2 with respect to u, we have − ∂ ∂z 1 ∂F ∂z = . (A8) ∂v ∂u −2 ∂z ∂u As a result, we ﬁnd

∂z z2 = C , (A9) ∂u v¯˙ where C is an integration constant which can be absorbed into normalization of the aﬃne parameter. Then, the “Killing energy” can be rewritten as ∂z ∂z E = C . (A10) − ∂v ∂u Since the boundary conditions at z = 0 are given by ∂z ∂z 1 = = , (A11) ∂v − ∂u −2 z=0 z=0

13 we have Eb = C, which is the energy observed at the boundary, and

E = E /λ. (A12) |z=z(s) b Using the geodesic equations, we also have

dE F ∂F v¯˙ 2 v¯˙ dF v¯˙ ∂F ∂ ln F = + = v¯˙ = Ev¯˙ . (A13) ds 2 ∂z 2z2 2z2 ds 2z2 ∂v¯ ∂v¯

Integrating it along the outgoing null geodesics described by z = zg(¯v), we obtain

v¯b ∂ ln F E = E exp d¯v , (A14) b i ∂v¯ Zv¯i z=zg(¯v) where Ei and Eb are the Killing energy observed at an initial surface and the boundary, respectively. Consequently, for the outgoing null ray described by u = const., we have the redshift factor from an initial timev ¯i to asymptotic time u =v ¯b as

u ∂ ln F λ(u) = exp d¯v . (A15) ∂v¯ Zv¯i z=zg(¯v) It turns out that if the spacetime is stationary, namely, F does not depend onv ¯, we have λ = 1. The redshift factor λ(u) deviates from the unity when time dependence is turned on.

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